Linear Discrepancy is Π_2-Hard to Approximate

07/02/2021

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by   Pasin Manurangsi, et al.

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In this note, we prove that the problem of computing the linear discrepancy of a given matrix is Π_2-hard, even to approximate within 9/8 - ϵ factor for any ϵ > 0. This strengthens the NP-hardness result of Li and Nikolov [ESA 2020] for the exact version of the problem, and answers a question posed by them. Furthermore, since Li and Nikolov showed that the problem is contained in Π_2, our result makes linear discrepancy another natural problem that is Π_2-complete (to approximate).